update text after edit mesh with handlers
The error NameError: name 'self' is not defined is pretty self explanatory, you are using self as a variable name and it has not been defined at the point you are using it. self is a variable name normally only used within a class definition while you are trying to use it in a module function.

self is normally defined to point to the object that the current function belongs to. In the context that you are trying to use it, it will be the first argument sent to the callback function, meaning it should point to the object that will be handling the callback.

I think to achieve what you want, try adding a variable to the scene object that holds a list of data that draw_text_callback draws (See the example for CollectionProperty), your update_scene maintains the list and draw_text_callback goes through it to draw what it needs to.

The second error is because you can't add another parameter to the function used for scene_update_post as it must match the single argument prototype that is defined to be used.